A SIMULATOR FOR TWENTY20 CRICKET JACK DAVIS, HARSHA PERERA AND TIM B. SWARTZ1 Simon Fraser University Summary This paper develops a Twenty20 cricket simulator for matches between sides belonging to the International Cricket Council. As input, the simulator requires the probabilities of batting out- comes which are dependent on the batsman, the bowler, the number of overs consumed and the number of wickets lost. The determination of batting probabilities is based on an amalgam of standard classical estimation techniques and a hierarchical empirical Bayes approach where the probabilities of batting outcomes borrow information from related scenarios. Initially, the probabilities of batting outcomes are obtained for the first innings. In the second innings, the target score obtained from the first innings affects the aggressiveness of batting during the second innings. We use the target score to modify batting probabilities in the second innings simulation. This gives rise to the suggestion that teams may not be adjusting their second innings batting aggressiveness in an optimal way. The adequacy of the simulator is addressed through various goodness-of-fit diagnostics. Keywords: Empirical Bayes, Markov chain Monte Carlo. 1Author to whom correspondence should be addressed. Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby BC, Canada V5A1S6 The authors wish to thank two anonymous reviewers for helpful comments that resulted in an improvement to the manuscript. Swartz has been supported by the Natural Sciences and Engineering Research Council of Canada. 1 1 INTRODUCTION The game of cricket has a long history dating back to the 16th century. The most recent form of cricket, known as Twenty20 cricket (or T20 cricket), began in 2003 involving matches between English and Welsh domestic sides. Since 2003, Twenty20 cricket has exploded in popularity with five World Cups having been contested (2007, 2009, 2010, 2012 and 2014). The Indian Premier League (IPL) which had its inaugural season in 2008 is known as the showcase for T20 cricket. The IPL continues to grow in popularity with respect to the number of teams, television contracts, salaries, etc. Except for some subtle differences (e.g. fielding restrictions, limits on the number of overs for bowlers, etc.), Twenty20 cricket shares many of the features of one-day cricket. One-day cricket was introduced in the 1960s, and like T20 cricket, it is a version of cricket based on limited overs. The main difference between T20 cricket and one-day cricket is that each batting side in T20 is allotted 20 overs compared to 50 overs in one-day cricket. This difference allows Twenty20 matches to finish in roughly three hours, a length of time comparable to the duration of matches in many other professional sports. Simulation methodologies have been developed and proven useful for many types of complex systems. For example, the simulation of weather systems using mathematical models has a long history in both short-term weather forecasts and in the prediction of climate change (Lynch 2008). A match simulator for T20 cricket would likewise be useful. For example, the prediction of match outcomes is obviously of interest to cricket enthusiasts. A match simulator would also facilitate the investigation of various match characteristics for which there does not exist a sufficient number of actual matches. For example, suppose that a T20 team is considering a new batting lineup. They may be interested in the distribution of runs scored by the hypothetical lineup. Naturally, a good simulator for T20 cricket is one which is realistic and captures the complexity of the game. To our knowledge, there have not been any realistic simulators developed for Twenty20 cricket. A difficulty in the development of a realistic T20 match simulator involves gaining a detailed understanding how various interacting factors (e.g. overs, wickets, batsmen, bowlers, the target, 2 the powerplay2, etc.) affect run progression. Simulators have been investigated for other forms of cricket. The earliest \simulators" were proposed by Elderton (1945) and Wood (1945) who fitted simple geometric distributions for the number of runs scored in test cricket. Dyte (1998) also considered the simulation of test cricket matches where the only inputs were career batting and bowling averages. In one-day cricket, Bailey & Clarke (2006) introduced covariates related to run scoring and used the normal distribution for the generation of runs. In test cricket, Scarf et al. (2011) model the number of runs by fitting a zero-inflated negative binomial distribution to each of the 10 partnerships. More closely related to this paper, Swartz, Gill & Muthukumarana (2009) developed a Bayesian latent variable model which provided batting outcome probabilities in one-day cricket. A criticism of Swartz, Gill & Muthukumarana (2009) is that they use a coarse discretization of wickets lost and overs consumed based on 9 overall categories. In particular, their structure does not account for powerplays. Section 2 is concerned with preliminaries related to the T20 simulator. We first introduce the extensive dataset which is used throughout the paper. Exploratory data analyses are carried out to motivate the subsequent modeling. The T20 simulator is then described in simple terms for first innings batting. Section 3 discusses the inputs to the simulator. Specifically, batting outcomes are enumerated and the corresponding probabilities are derived from multinomial distributions. Our model is highly parametrized and we use an amalgam of classical estimation techniques and a hierarchical Bayesian model to estimate the multinomial parameters. One of the key features of the approach is that the estimators from a given scenario borrow information from related scenarios to improve reliability. Another noteworthy aspect of the approach concerns the detail which is provided in ball-by-ball scoring. Simulators which simply generate the total number of runs for each over do not address the manner in which runs are scored. In section 4, the simulator is extended in various ways. We consider the case of specific batsman/bowler matchups, the home team advantage and second innings simulation where the target score is taken into account. Clearly, higher target scores force the second innings batting team to be more aggressive. When they are more aggressive, they score more runs but are more likely to be dismissed. In section 2the powerplay is defined later 3 5, we demonstrate the realism of the simulator via some goodness-of-fit diagnostics. A notable consequence of the validation exercise is the suggestion that teams may not be batting optimally during the second innings. Specifically, teams that are falling behind in the second innings may not be increasing their aggressiveness in an incremental fashion. We also illustrate the utility of the simulator by addressing some problems of prediction. We conclude with a short discussion in section 6. 2 PRELIMINARIES For the analysis, we consider all T20 matches that took place from 2005 until the end of 2013 which involved full member nations of the International Cricket Council (ICC). Currently, the 10 full members of the ICC are Australia, Bangladesh, England, India, New Zealand, Pakistan, South Africa, Sri Lanka, West Indies and Zimbabwe. Details from these matches can be found in the Archive section of the CricInfo website (www.espncricinfo.com). A proprietary R-script was used to parse and extract ball-by-ball information from the Match Commentaries. In total, we obtained data from 250 matches. In Table 1, we provide summary statistics for the matches where we observe that Bangladesh and Zimbabwe are clearly the weakest T20 teams. Amongst the other 8 teams, the winning percentages do not vary greatly. When looking at the differences between runs scored versus runs allowed for individual teams, it appears that Sri Lanka's win percentage is lower than what might be expected. We now study various features related to batting. We temporarily ignore extras (sundries) that arise via wide-balls and no-balls, and note that there are only 8 broadly defined outcomes that can occur when a batsman faces a bowled ball. These batting outcomes are listed below: outcome j = 0 ≡ 0 runs scored outcome j = 1 ≡ 1 runs scored outcome j = 2 ≡ 2 runs scored outcome j = 3 ≡ 3 runs scored (1) outcome j = 4 ≡ 4 runs scored outcome j = 5 ≡ 5 runs scored outcome j = 6 ≡ 6 runs scored outcome j = 7 ≡ dismissal 4 Team Matches Win % R¯(S) R¯(A) Australia 63 52% 161.7 (33) 157.8 (30) Bangladesh 22 18% 134.0 (07) 169.5 (15) England 58 55% 159.7 (23) 158.7 (36) India 42 50% 159.4 (23) 164.7 (19) New Zealand 64 44% 153.6 (32) 157.8 (32) Pakistan 67 55% 152.3 (36) 144.9 (31) South Africa 58 62% 147.6 (29) 148.9 (29) Sri Lanka 52 48% 159.7 (26) 139.3 (26) West Indies 49 41% 159.7 (28) 147.5 (21) Zimbabwe 25 20% 129.9 (13) 171.8 (12) Table 1: Summary statistics for the T20 dataset corresponding to matches from February 17, 2005 through November 13, 2013. The variables R¯(S) and R¯(A) denote the average number of first innings runs scored and runs allowed, respectively, with the number of matches in parentheses. In the list (1) of possible batting outcomes, we include byes, leg byes and no balls where the resultant number of runs determines one of the outcomes j = 0;:::; 7. We note that the outcome j = 5 is rare but is retained to facilitate straightforward notation. We first calculate the proportionsp ^0;:::; p^7 corresponding to the first innings batting out- comes. Table 2 provides a comparison of these proportions based on the T20 dataset compared with the proportions for fourth innings batting in test cricket as reported by Perera, Gill & Swartz (2013). We observe that T20 batting is much more aggressive than batting in test cricket.